접수일자 : 2010년 4월 14일
Fuzzy r-minimal Preopen Sets And Fuzzy r- M Precontinuous Mappings On Fuzzy Minimal Spaces
민원근*․김영기**
Won Keun Min and Young Key Kim
* 강원대학교 수학과
** 명지대학교 수학과
요 약
fuzzy r-minimal preopen set, fuzzy r-M precontinuous와 fuzzy r-M preopen 함수의 개념을 소개하며 특성들을 조사 한다.
Abstract
We introduce the concept of fuzzy r-minimal preopen set on a fuzzy minimal space. We also introduce the concept of fuzzy r-M precontinuous mapping which is a generalization of fuzzy r-M continuous mapping, and investigate characterization of fuzzy r-M precontinuity.
Key Words: fuzzy minimal structures, r-minimal open, r-minimal preopen, fuzzy r-M continuous, fuzzy r-M precontinuous
1. Intorduction
The concept of fuzzy set was introduced by Zadeh [5]. Chang [1] defined fuzzy topological spaces using fuzzy sets. In [2], Chattopadhyay, Hazra and Samanta introduced the smooth topological space which is a generalization of a fuzzy topological space. In [4], we introduced the concept of fuzzy -minimal space which is an extension of the smooth topological space. The concepts of fuzzy -open sets and fuzzy -M con- tinuous mappings are also introduced and studied.
In this paper, we introduce the concept of fuzzy -minimal preopen sets and study some related properties. We also introduce the concept of fuzzy -M precontinuous mappings, which is a generalization of fuzzy -M continuous mapping. In particular, we in- vestigate some characterization for the fuzzy -M pre- continuous mapping in terms of fuzzy -minimal in- terior and fuzzy -minimal closure operators.
2. Preliminaries
Let be the unit interval of the real line. A member of is called afuzzy set [5] of . By
and , we denote constant maps on X with value and
, respectively.
For any ∈, denotes the complement -. All other notations are standard notations of fuzzy set theory.
A fuzzy point in is a fuzzy set is defined as follows
=
i f i f ≠
A fuzzy point is said to belong to a fuzzy set in , denoted by ∈, if ≤ for ∈.
A fuzzy set in is the union of all fuzzy points which belong to .
Let → be a mapping and ∈ and ∈. Then is a fuzzy set in , defined by
∈
i f ≠ ∅
otherwise
for ∈ and is a fuzzy set in , defined by
∈.
A smooth topology[2, 3] on is a map → which satisfies the following properties:
(1) ()=()=1.
(2) (∧)≥()∧() for , ∈. (3) (∨)≥∧() for ∈.
The pair is alled a smooth topological space.
Let be a nonempty set and ∈(0,1]=. A fuzzy family → on X is said to have a fuzzy -minimal structure [4] if the family
={∈ : ()≥}
contains and .
Then the (,) is called a fuzzy -minimal space (simply -FMS) [4]. Every member of is called a fuzzy -minimal open set. A fuzzy set is called a fuzzy -minimal closed set if the complement of (simply, ) is a fuzzy -minimal open set.
Let (,) be an -FMS and ∈(0,1]=. The fuzzy
-minimal closure of , denoted by mC(,), is defined as
mC(,)=∩{∈: ∈ and ⊆}[4].
The fuzzy -minimal interior of , denoted by mI(,
), is defined as
mI(,)=∪{∈: ∈ and ⊆}[4].
Theorem 2.1 ([4]). Let (,) be an -FMSand ,
∈. Then the following properties hold:
(1) mI(,)⊆ and if A is a fuzzy -minimal open set, then mI(,)=.
(2) ⊆mC(,) and if is a fuzzy -minimal closed set, then mC(,)=.
(3) If ⊆, then mI(,)⊆mI(,) and mC(,)⊆
mC(,).
(4) mI(∩,)⊆mI(,)∩mI(,) and mC(,)∪
mC(,)⊆mC(∪,).
(5) mI(mI(,),)=mI(,) and mC(mC(,),)= mC (,).
(6) -mC(,)=mI(-,) and -mI(,)=mC(-,
).
Definition 2.2 ([4]). Let (,) and (,) be -FMS's. Then → is said to be fuzzy -M con- tinuous function if for every ∈, is in .
Theorem 2.3 ([4]). Let → be a function on -FMS's (,) and (,).
(1) is fuzzy -M continuous.
(2) is a fuzzy -minimal closed set, for each fuzzy -minimal closed set in .
(3) (mC(,))⊆mC((),) for ∈. (4) mC( (),)⊆ (mC(,)) for ∈. (5) (mI(,))⊆mI( (),) for ∈. Then (1) ⇔ (2) ⇒ (3) ⇔ (4) ⇔ (5).
3. Fuzzy r-minmal preopen sets and fuzzy r- M precontinuity
Definition 3.1. Let (,) be an -FMS and ∈
. Then a fuzzy set is called a fuzzy -minimal preopen set in if ⊆mI(mC( ),).
A fuzzy set is called a fuzzy -minimal spre- closed set if the complement of is fuzzy -minimal preopen.
Every fuzzy -minimal open set is fuzzy -minimal preopen but the converse may not be true in general.
Example 3.2 Let = , let , and be fuzzy sets defined as follows
∈
∈
∈
Let us consider a fuzzy minimal structure M as fol- lows
i f
Then the fuzzy set is fuzzy
-minimal preopen
but not fuzzy
-minimal open.
Lemma 3.3. Let (,) be an -FMS and ∈. Then a fuzzy set is fuzzy -minimal preclosed set if and only if mC(mI(,),)⊆.
Theorem 3.4. Let (,) be an -FMS. Any union of fuzzy -minimal preopen sets is fuzzy -minimal preopen.
Proof. Let be a fuzzy -minimal preopen set for
∈. Then from Theorem 2.1, ⊆mI(mC(,),) ⊆ mI(mC(∪,),). This implies ∪⊆mI(mC(∪,),
) and so ∪ is fuzzy -minimal preopen.
As shown in the next example, in general, the inter- section of two fuzzy -minimal preopen sets may not be fuzzy -minimal preopen.
Example 3.5. Let = , let , , and , be fuzzy sets defined as follows
i f ∈
i f ∈
i f ≤ ≤
i f
≤ ≤
i f ≤ ≤
i f
≤ ≤
Let us consider a fuzzy minimal structure
i f
Then the fuzzy sets and are fuzzy
-minimal
preopen. But ∩ is not fuzzy
-minimal preopen,
because of mI(mC(∩,
),
)=∪⊆, but
∪≠, .
Definition 3.6. Let (,) be an -FMS and ∈
, mpC( ) and mpI( ), respectively, are defined as the following:
mpC( )=∩{∈: ⊆ and is fuzzy
-minimal preclosed},
mpI( )=∪{∈: ⊆ and is fuzzy
-minimal preopen}.
Theorem 3.7. Let (,) be an -FMS and ∈. Then(1) mpI(,)⊆.
(2) If ⊆, then mpI(,)⊆mpI(,).
(3) is fuzzy -minimal preopen if and only if mpI (,)=.
(4) mpI(mpI(,),)=mpI(,).
(5) mpC(-,)=-mpI(,) and mpI(-,)=
-mpC(,).
Proof. (1), (2), (3) and (4) are obviously obtained from Theorem 3.4.
(5) For ∈,
-mpI(,)
=-∪{∈:⊆ and is fuzzy -minimal preopen}
=∩{- : ⊆, is fuzzy -minimal
=∩{- : -⊆-, is fuzzy -minimal preopen}
= mpC(-,).
Similarly, mpI(-,)=-mpC(,).
Theorem 3.8. Let (,) be an -FMS and ∈. Then(1) ⊆mpC(,).
(2) If ⊆, then mpC(,)⊆mpC(,).
(3) is fuzzy -minimal preclosed if and only if mpC()=.
(4) mpC(mpC(,),)=mpC(,).
Proof. It is obvious from Theorem 3.7.
Lemma 3.9. Let (,) be an -FMS and ∈. Then(1) ∈mpC(,) if and only if ∩≠ ∅ for ev-
ery fuzzy -minimal preopen set V containing . (2) ∈mpI(,) if and only if there exists a fuzzy
-minimal preopen set such that ⊆.
Proof. (1) If there is a fuzzy -minimal preopen set
containing such that ∩ =∅, then - is a fuzzy -minimal preclosed set such that ⊆-,
∉-. So ∉mpC(,).
The other relation is obvious.
(2) Obvious.
Definition 3.10. Let (,) and (,) be -FMS's.
Then → is said to be fuzzy -M preicontinuous if for each fuzzy point and each fuzzy -minimal open set containing (), there exists a fuzzy -minimal preopen set containing such that ()
⊆.
Every fuzzy -M continuous mapping is fuzzy -M precontinuous but the converse is not true in general.
Example 3.11. Let = . Consider two fuzzy minimal structures (,) and (,) defined in Example 3.2 and Example 3.5, respectively. Then the identity mapping → is fuzzy
-
precontinuous but not fuzzy
- continuous.
Theorem 3.12. Let → be a mapping on fuz- zy -minimal spaces (,) and (,). Then the fol- lowing are equivalent:
(1) is fuzzy -M precontinuous.
(2) () is a fuzzy -minimal precopen set for
(3) () is a fuzzy -minimal precclosed set for each fuzzy -minimal closed set in .
(4) (mpC(,))⊆mC((),) for ∈. (5) mpC( (),)⊆ (mC(,)) for ∈. (6) (mI(,))⊆mpI( (),) for ∈. Proof. (1) ⇒ (2) Let be any fuzzy -minimal open set in and ∈ (). Then there exists a fuzzy -minimal precopen set containing such that ()
⊆ and so ⊆ () for all ∈ (). Hence from Theorem 3.4, () is fuzzy -minimal precopen.
(2) ⇒ (3) Obvious.
(3) ⇒ (4) For ∈,
(mC((,)))
= (∩{∈: ()⊆ and is a fuzzy
-minimal closed set})
=∩{ ()∈: ⊆ () and is a fuzzy
-minimal precclosed set}
⊇ ∩{∈: ⊆ and is a fuzzy -minimal precclosed set}
=mpC(,)
Therefore, (mpC(,))⊆mC((),).
(4) ⇒ (5) For ∈,
(mpC( (),))⊆mC(( ()),)⊆mC(,).
Thus this implies mpC( (),)⊆ (mC(,)).
(5) ⇒ (6) For ∈, from Theorem 3.7,
(mI(,))= (-mC(-,))
=-( (mC(-),))
⊆-mpC( (-))
=mpI( (),).
Hence (mI(,))⊆mpI( (),).
(6) ⇒ (1) Let be any fuzzy -minimal open set containing () for a fuzzy point . From (6), it fol- lows
∈ ()= (mI(,))⊆mpI( (),).
Since ∈mpI( (),), there exists a fuzzy -minimal precopen set containing such that ⊆
(). Hence is fuzzy -M preccontinuous.
Lemma 3.13. Let (,) be an -FMS and ∈. Then
(1) mC(mI(,),)⊆mC(mI(mpC(,),),)⊆mpC(,).
(2) mpI(,)⊆mI(mC(mpI(,),),)⊆mI(mC(,),).
Proof. (1) Since ⊆mpC(,) and mpC(,) is fuz- zy -minimal preclosed, it is obtained from Lemma 3.3 and Theorem 3.8.
(2) Similarly, it follows from Theorem 3.7.
Theorem 3.14. Let → be a mapping on -FMS's (,) and (,). Then the following state- ments are equivalent:
(1) is fuzzy - precontinuous.
(2) ()⊆mI(mC( (),),) for each fuzzy -minimal open set in .
(3) mC(mI( (),),)⊆ () for each fuzzy -minimal closed set in .
(4) (mC(mI(,),))⊆mC((),) for ∈. (5) mC(mI( (),),)⊆ (mC(,r)) for ∈. (6) (mI(,))⊆ mI(mC( (),),) for ∈. Proof. (1) ⇔ (2) It is obtained from definition of fuz- zy -minimal preopen sets and Theorem 3.12.
(1) ⇔ (3) It follows from Lemma 3.3 and Theorem 3.12.
(3) ⇒ (4) Let ∈. Then from Lemma 3.13(1) and hypothesis, it follows mC(mI(,),)⊆mpC(,)
⊆ ((mpC(,)))⊆ (mC((),)).
Hence (mC(mI(,),))⊆ mC((),).
(4) ⇒ (5) Obvious.
(5) ⇒ (6) Let ∈. Then from Theorem 2.1,
(mI(,))= (-mC(-,))
=-( (mC(-,)))
⊆-mC(mI( (-),),)
=mI(mC( (),),).
Hence (6) is obtained.
(6) ⇒ (1) Let be any fuzzy -minimal open set.
Then since =mI(,), by (6), we have ()= (mI(,))⊆mI(mC( (),),). By Theorem 3.7 (3),
() is fuzzy -minimal preopen and so is fuzzy - precontinuous.
References
[1] C L. Chang, "Fuzzy topological spaces", J. Math.
Anal. Appl., vol. 24, pp. 182-190, 1968.
[2] K. C. Chattopadhyay, R. N. Hazra, and S. K.
Samanta, "Gradation of openness: Fuzzy top- ology", Fuzzy Sets and Systems, vol. 49, pp.
237-242, 1992.
[3] A. A. Ramadan, "Smooth topological spaces", Fuzzy Sets and Systems, vol. 48, pp. 371-375, 1992.
[4] Y. H. Yoo, W. K. Min and J. I. Kim. "Fuzzy -Minimal Structures and Fuzzy -Minimal Spaces", Far East Journal of Mathematical Sciences, vol. 33, no. 2, pp.193-205, 2009.
[5] L. A. Zadeh, "Fuzzy sets", Information and Control, pp. 338-353, 1965.
저 자 소 개
Won Keun Min
Department of Mathematics, Kangwon National University, Chuncheon, 200-701, Korea
Young Key Kim
Department of Mathematics, MyongJi University, Youngin 449-728, Korea.
